Konta and Watson both won their singles matches as Britain saw off Portugal 3-0 on Wednesday, and the pair were triumphant again on Friday against host nation Estonia in Tallinn.
After Watson had thrashed 15-year-old Katriin Saar 6-1 6-1 in just 57 minutes first up, Konta eventually emerged victorious in a much-tougher encounter with world number 27 Anett Kontaveit to earn Britain an unassailable 2-0 lead in the tie.
Konta served twice for the first set but ended up losing it on a tie-break as Estonia, who also beat Portugal in their opening match, drew first blood.
However, world number 11 Konta hit back and eventually clinched a 6-7 (5/7) 6-4 6-2 victory to send Britain through to a play-off, where they will face the winner of Group C.
Johanna Konta of Great Britain shakes hands with Great Britain team
Image credit: Getty Images
If Anne Keothavong’s team win that tie then they qualify for the World Group II play-offs in April.
Britain have made it to the World Group II play-offs three times in the last six years – including last year against Romania – but were beaten on each occasion.
Britain have not played in the Fed Cup World Group since 1993.